The book (that you didn't originally identify in the question) mentions NOTHING of anything called "Free People of Earth". Neither the Ender's Game you nominated, nor even Ender's Shadow with its more geocentric details. Maybe it's from later within the Shadows Saga, but i wouldn't count on that.
[Later Edit:] After phantom42 mentioned which book actually tells the story of the Free People, i went over the whole of the series and re-checked them all, not just the two i believed you referred to by using the tag of "enders-game"... So here's the scoop:
- the second half of SotG mentions how FPoE started and how it evolved until "the most belligerent nations had been subdued";
- the final note on the subject was: "Only the good old USA refuses to join the FPE" (note that this is the official shortened name, not FPOE). There's no evidence of what happened next, only Peter's guess that a peaceful union would soon be possible. Note that at this point, i believe a war between US and the Hegemony wouldn't be feasible.
- the currently-latest novel, Shadows in Flight, while being the chronological sequel, is no longer set on Earth and (in its abridged e-book version) doesn't reveal any details about the aftermath of FPE's "taking over";
- for completeness' sake, neither of the three Lusitania volumes mention FPE at all, which makes sense considering they were published before the events of the "Shadow"s would bring forth this part of the story.